# Differential Equation satisfied by Weierstrass's Elliptic Function

## Theorem

$\paren {\dfrac {\d f} {\d z} }^2 = 4 f^3 - g_2 f - g_3$

where:

$\ds g_2 = 60 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^4}$

and:

$\ds g_3 = 140 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^6}$

has the general solution:

$\map f z = \map \wp {z + C; \omega_1, \omega_2}$

where:

$\wp$ is Weierstrass's elliptic function
$C$ is an arbitrary constant
$\omega_1$, $\omega_2$ are constants independent of $z$.